Math 263C Spring 2012 Final Exam Sample 2
Write your solutions on the blank paper provided. Show all your work. Circle or box your
nal answer. Each problem carries equal weight.
Instructors Last Name:
1. If a =< 1, 1, 3 > and b =< 5, 3, 2 > nd the followi
Math 263c - Quiz 3 Information
General Information
Date: Monday, May 14th
Length: 10 minutes
No calculators or reference material will be allowed.
Topics Included
1. Section 10.1 Three-Dimensional Coordinate Systems
(a) sketch lines, line segments, and
Math 263c Quiz 4
Due: Tuesday May 22, at the end of class.
This quiz will work just like a group work, except that it is due by the end of class, and I
will record the score as a quiz grade, instead of a homework grade. Work in groups of 2-3
and turn them
Math 263c Worksheet 10.8 Arc Length and Curvature
Due: Friday May 25, at the end of class.
Work on these problems in groups of 2-3 and turn them in together.
1. Find the length of the curve r(t) =< 12t, 8t3/2 , 3t2 >,
0 t 1.
2. Let r(t) =< t, 0.5t2 , t2 >
MATH 263C, Winter 2007, SOLUTION of Midterm 2
1. Find the radius and interval convergence of the series:
(a) (5 points)
1
P
n=1
(1)n1 xn
.
n3
n+1
3
3
x
n
Answer. We use the Limit Ratio Test. lim j an+1 j = lim j (n+1)3 : xn j = lim jxj (n+1) =
an
n3
3
2
+
.
Math 263C, WINTER 2007, SOLUTION of Quiz
Show all your work to get credit. No work will amount to no credit.
1
(a) Evaluate the value of the series
1
n=2
1
n2 1
Answer: Since n2 1 = (n 1)(n + 1) we have
Hence
1
n=2
n2
1
n2 1
1
= 1 [ n1
2
1
n+1 ].
1
1
1
.
Math 263C, WINTER 2007, SOLUTION of Quiz 2
Question: Are the following series convergent or divergent? Why?
2
(a) 1 cosn(n )
3
n=1
2
1
Answer: This series is convergent, because 1 cosn(n ) = 1 n3 is a p-series
3
n=1
n=1
with p = 3, convergent.
n+2
(b) 1
Math 263C, WINTER 2007, Solution of Quiz 3
1. Find the radius of convergence and the interval of convergence of the series
xn
.
n3n
n=1
ANSWER: To nd radius and interval of convergence we need to use the Limit
Ratio Test:
an+1
lim
= lim
n
n
an
xn+1
(n+1)3
MATH 263C, Winter 2007, Solution of HW of 11.9
The assigned HW of 11.9 is 1, 2, 5-29 odd numbers. Here are solutions of 5-29
(odd). As in the class, I called the following power series "a basic series" and denoted
it by (*), for easy reference, as follows
.
Math 263C, WINTER 2007, Solution of HW 11.10
Attn. If f (x) has a power series representation at a, this means, if
f (x) =
1
X
n=0
cn (x a)n ; jx aj < R
then its coecients are given by
cn =
f (n) (a)
:
n!
9. Find a power series representation for the fu
MATH 263C, Winter 2007, SOLUTION of HW 12.4
HW numbers 25, and 33 of 12.4 were already discussed in the class. Here are
solutions of number 15, number 29 and number 31 for you to study. You have to do
also problems 1-13 (odd), 23, 27. But these problems a
Math 263C Spring 2012 Final Exam Sample 1
Write your solutions on the blank paper provided. Show all your work. Circle or box your
nal answer. Each problem carries equal weight.
Instructors Last Name:
1. If a =< 1, 3, 5 > and b =< 2, 4, 6 > nd the followi
Math 263C Spring 2012 Midterm Exam
Write your solutions on the blank paper provided. Show all your work. Circle or box your
nal answer. Each problem carries equal weight. When you are nished, put your solutions
in order, put the question sheet on top, and
Math 263c Worksheet 8.5 Power Series Due: Monday April 16th Work on these problems in groups of 2-3 and turn them in together.
1. Find the radius of convergence and interval of convergence of the series. (a) (b) (c) (d) (e) (f) 2. If xn n7n xn n10 10n n!(
Math263C - Optional Assignment
Due: Monday April 30th By the end of the quarter we will have accumulated approximately five quizzes and seven homework/groupwork assignments. Since quizzes are a total of 10% of the grade, then each quiz will be worth 2 per
Math 263C Spring 2012 Midterm Exam Write your solutions on the blank paper provided. Show all your work. Circle or box your final answer. Each problem carries equal weight. When you are finished, put your solutions in order, put the question sheet on top,
Math 263c Homework Rubric When working on homework assignments there should be a balance of working alone and working in a group. You should first attempt the assignment by yourself and then meet with a group to discuss any difficulties or to compare solu
Math 263c - Quiz 1 Information
General Information Date: Tuesday, April 10th Length: 15 minutes No calculators or reference material will be allowed. Topics Included 1. Section 4.6 Newton's Method (a) how it works (b) several downfalls 2. Section 8.1 Sequ
Math 263c - Quiz 1 Information
General Information Date: Monday, April 23rd Length: around 15-20 minutes No calculators or reference material will be allowed. Topics Included 1. Section 8.4 Other Convergence Tests (a) ratio test (b) root test (c) absolute
Taylor approximations of the cosine function
Maclaurin series for cosine
Below is an animation showing the nth degree Taylor polynomials (n = 2, 4, 6, 8, 10) of the function cos x about 0, and how they approximate the curve y = cos x. Notice that as n inc
Taylor approximations of the exponential function
Maclaurin series for ex
Below is an animation showing the nth degree Taylor polynomials (n = 1, . . . , 5) of the function ex about 0, and how they approximate the curve y = ex . Notice that as n increases
Taylor approximations of geometric function
Maclaurin series for 1 1-x
1 1-x
Below is an animation showing the nth degree Taylor polynomials (n = 1, . . . , 9) of the function about 0, and how they approximate the curve y =
1 . Notice that as n increases
Math 263c Worksheet 8.1 Sequences Due: Thursday March 29th 1. Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. (a) cfw_ 1/4, 2/9, 3/16, 4/25, . . . (b) cfw_ -2, 4/2, -8/6, 16/24, . . . 2
Math 263c Worksheet 8.2 & 8.3 Series and The Integral Test Due: Thursday April 5th Work on these problems in groups of 2-3 and turn them in together.
1. What name is given to the following series? For which values of b does each series converge?
(a)
n=1
MATH 263C, Winter 2007, Midterm 1
Name: SOLUTION
1. (5 points) Find a series 1 an with limn!1 an = 0 but the series is divergent.
n=1
1
1
ANSWER: The series 1 n or 1 pn have this property, because they are
n=1
n=1
divergent as p-series with p less than or